There was an observation that any algebraic curve over Q can be rationally mapped to P^1 without three points and this led Grothendieck to define a special class of these mappings, called the Children's Drawings, or, in French, Dessins d'Enfants (his quote was something like "things as simple as the drawings...").

I'm not an expert in this field, so could somebody please write more about those dessins, and what things they are related to? What's their importance? How does the cartographic group act on these?

Is there any way to see in any geometric way the shuffle relations (satisfied by multi-zeta values) or the "automorphisms of the associator" (arising from Drienfeld's work on quasi-Hopf algebras) that appear in the larger body of work related to dessins?
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Dr ShelloMay 2 '11 at 22:47

In Leila Schneps - Dessins d'enfants on the Riemann Sphere you can find a definition of dessins and the Grothendieck correspondence between Belyi morphisms and dessins. It also has pictures of how the cartographic group acts on the flag set of a dessin.
Grothendieck correspondence means that there is a bijection between isomorphism classes of dessins and isomorphism classes of Belyi morphisms (morphisms f:X->P1C which are ramified only over three points).

Historically, one of the first papers on th subject is Drawing curves over number fields, by G.B. Shabat and V.A. Voevodsky (The Grothendieck Festschrift, Vol. III, 199–227,
Progr. Math., 88), which I strongly recommend. Another nice and historical paper is Triangulations, by M. Bauer and C. Itzykson (many references: R.C.P. 25, Vol. 44 (1992), Discrete Math. 156 (1996), no. 1-3 or L. Schneps's book below). Both papers are also concerned with the (combinatorial) cellular decomposition of moduli spaces of curves. They appeared before L. Schneps book The Grothendieck theory of dessins d'enfants (London Mathematical Society Lecture Note Series, 200), which is now the main reference on the subject.

There are many good answers to this question already. However it seems important to me that the contribution of Jones and Singerman to this subject is noted. These two British mathematicians from the University of Southampton wrote an important paper on this subject some time before Grothendieck wrote his Esquisse.

The paper is beautifully written, and outlines the correspondence between maps on topological surfaces, maps on Riemann surfaces, and groups with certain distinguished generators. They do not consider the Galois action, this being the aspect of the area that so excited Grothendieck. Their notion of a map is a particular instance of a dessin d'enfant (these days a map is also known as a clean dessin), the more general notion of hypermap which was considered subsequently corresponds to the general dessin d'enfant.

A later paper, by Bryant and Singerman, extended the treatment to surfaces with boundary.

Grothendieck's dessins d'enfants are closely connected to the study of coverings of the three
times punctured sphere, and such coverings can be considered from many different points of view.
In this survey it is shown how all of them are equivalent, and how the absolute
Galois group acts on these objects.

From the text (translated from the French): "In 1984, A. Grothendieck presented a research program,
entitled Esquisse d'un programme' (published in 1997 [in Geometric Galois actions, 1, 5--48,
Cambridge Univ. Press, Cambridge, 1997; MR1483107 (99c:14034)]), as part of his application for
a position at the CNRS (a position that he would hold until his retirement in 1988). In his program Grothendieck
used the termdessin d'enfant' (in its ordinary sense) as a visual analogue of certain cell maps;
he explained that every finite oriented map is realized canonically over a complex algebraic curve' and that
the Galois group of $\overline{\bf Q}$ over $\bf Q$ acts on the category of these maps in a natural way':
one derives this by comparing various approaches to the study of coverings of $\bf P_1 - \{0,1,\infty\}$.
Since then, the term `dessin d'enfant' has been used often, by various authors in various mathematical senses,
to denote objects (or isomorphism classes of objects) arising in those approaches.
In this paper we do not try to define the term; we content ourselves with using it to denote the theory as a whole.

"Here are some reasons why one should pay particular attention to finite coverings of the
curve $\bf P_1 - \{0,1,\infty\}$:
"(a) It is the simplest algebraic curve whose fundamental group is not commutative.
"(b) It has many coverings over $\overline{\bf Q}$: according to a theorem of Belyi(, every
integral algebraic curve over $\overline{\bf Q}$ has an open Zariski set that is realized as such a covering.
"(c) It is identified with the moduli space $M_{0,4}$ of genus-0 curves equipped with four
marked points. The study of the action of ${\rm Gal}(\overline{\bf Q}/\bf Q)$ on its $\pi_1$
is the starting point for the study of the Grothendieck-Teichmüller tower
(consisting of the fundamental groupoids of all the moduli spaces
$M_{g,n}$ on which ${\rm Gal}(\overline{\bf Q}/\bf Q)$ acts).''